Explanation Formula used: The procedure to solve the non-homogeneous system x ′ ( t ) = A x ( t ) + f ( t ) : Step-1: Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues: Step-2: Apply the | A − λ I | X = 0 in order to calculate eigenvectors u 1 and u 2 corresponding to eigenvalue λ 1 and λ 2 respectively: Step-3: Write the general solution of system x ′ ( t ) = A x ( t ) as: X ( t ) = c 1 e λ 1 t u 1 + c 2 e λ 2 t u 2 Step-4: Let X 1 fundamental matrix for the homogeneous system. A particular solution to the non-homogeneous system is given by the variation of parameter formula X p ( t ) = X 1 ∫ X 1 − 1 f ( t ) d t Calculation: Consider the system x ′ ( t ) = A x ( t ) + f ( t ) , A = [ − 4 2 2 − 1 ] , f ( t ) = [ e 4 t 3 e 4 t ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues: | [ − 4 2 2 − 1 ] − λ [ 1 0 0 1 ] | = 0 | − 4 − λ 2 2 − 1 − λ | = 0 Expand the above determinant: ( 4 + λ ) ( 1 + λ ) − 4 = 0 4 + 5 λ + λ 2 − 4 = 0 λ ( λ + 5 ) = 0 Equate the both product to zero: λ = 0 or ( λ + 5 ) = 0 λ = 0 or λ = − 5 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 0 : [ − 4 2 2 − 1 ] [ x 1 x 2 ] = [ 0 0 ] Write above matrix notation in scalar form notation: − 4 x 1 + 2 x 2 = 0 2 x 1 − x 2 = 0 The eigenvector corresponding to eigenvalue λ = 0 : u 1 = [ 1 2 ] Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = − 5 : [ − 4 + 5 2 2 − 1 + 5 ] [ x 1 x 2 ] = [ 0 0 ] [ 1 2 2 4 ] [ x 1 x 2 ] = [ 0 0 ] Write above matrix notation in scalar form notation: x 1 + 2 x 2 = 0 2 x 1 + 4 x 2 = 0 The eigenvector corresponding to eigenvalue λ = 3 : u 2 = [ 2 − 1 ] The general solution of homogeneous system corresponding to eigenvector u 1 and u 2 : X h ( t ) = c 1 [ 1 2 ] + c 2 e − 5 t [ 2 − 1 ] The fundamental matrix for above general solution: X 1 = [ 1 2 e − 5 t 2 − e − 5 t ] Find the determinant of matrix:

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 9.RP, Problem 8RP

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To solve: The system x′(t)=Ax(t)+f(t), where A and f(t) are given

A=[−422−1], f(t)=[e4t3e4t].

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Chapter 9.RP, Problem 7RP

Chapter 9.RP, Problem 9RP

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Chapter 9 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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The system x ′ ( t ) = A x ( t ) + f ( t ) , where A and f ( t ) are given A = [ − 4 2 2 − 1 ] , f ( t ) = [ e 4 t 3 e 4 t ] .

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To solve: The system x′(t)=Ax(t)+f(t), where A and f(t) are given A=[−422−1], f(t)=[e4t3e4t].

Want to see the full answer?

Chapter 9 Solutions

To solve: The system x′(t)=Ax(t)+f(t), where A and f(t) are given

A=[−422−1], f(t)=[e4t3e4t].